I have found it depends on the zipline operator you are dealing with. In St Thomas, I used my camera case and attached it to me with a big caliper clip and a strap. It held my camera, key card, ID, credit card, and some cash along with my small camera. They provided us with sleeves for water bottles that they attached to our zipline harnesses. We used plenty of sunscreen before we left the ship which worked fine for us. I believe they also have somewhere you can lock things up if you need to.
Most places we have been ziplining do allow you to take a small bag as long as it is firmly attached to you. Remember, if you drop something, you may never see it again! Pockets that zip or button securely closed can also be a good way to bring the essentials with you. However, we have found some tour operators that do not let you bring much of anything with you. Those places pretty much always do provide secure locker space for you to leave your belongings. We went to Fly High ziplines in Cozumel on our last cruise and they were one of the places that didn't let us bring anything, but our things 8were safe in the lockers. We did keep our ID's, money, etc in our pockets there and had a great time on their ziplines! We loved Tree Limin in St Thomas, too!